Chapter 6: Product operators - The James Keeler Group

ijijDQ ⎯→⎯ cosπJ t DQ + cosπJ t 2IDQijijDQ ⎯→⎯cosπJ t DQ −sinπJ t 2IDQijijZQ ⎯→⎯ cosπJ tZQ + sinπJ t 2IZQZQ( ) ( ) ( ij)xDQ,eff xDQ,eff kz y( ) ( ) ( ij)yDQ,eff yDQ,eff kz x( ) ( ) ( ij)xZQ,eff xZQ,eff kz y( ij)( ij) ( ij)y⎯→⎯cosπJZQ,efftZQy− sinπJZQ,efft 2IkzZQxJ DQ,effis the sum of the couplings between spin i and all other spins plus the sumof the couplings between spin j and all other spins. J ZQ,effis the sum of thecouplings between spin i and all other spins minus the sum of the couplingsbetween spin j and all other spins.For example in a three-spin system the zero-quantum coherence betweenspins 1 and 2, anti-phase with respect to spin 3, evolves according to( 12) ( 12) ( 12)2I3zZQ y⎯→⎯cosπJZQ,efft 2I3zZQ y−sinπJZQ,efftZQxwhere J = J − JZQ,eff13 23Further details of multiple-quantum evolution can be found in section 5.3 ofErnst, Bodenhausen and Wokaun Principles of NMR in One and TwoDimensions (Oxford University Press, 1987).6–16